GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS

Yongqi Liu; Qigui Yang

doi:10.11948/20190106

2020 Volume 10 Issue 1

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2020 > 10(1): 192-209

Next Article Previous Article

Yongqi Liu, Qigui Yang. GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 192-209. doi: 10.11948/20190106

Citation:

Yongqi Liu, Qigui Yang. GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 192-209. doi: 10.11948/20190106

GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS

Yongqi Liu^1, ,,
Qigui Yang²

1.
College of Applied Mathematics, Beijing Normal University, Zhuhai, 519087 Guangdong, China
2.
Department of Mathematics, South China University of Technology, 510640 Guangzhou, China

Corresponding author: Email address: liuyongqi1980@163.com (Y. Liu)
Fund Project: The authors were supported by National Natural Science Foundation of China (No. 11871108) and Teacher Research Capacity Promotion Program of Beijing Normal University Zhuhai

Abstract

Abstract

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.
- HIV model /
- Global stability analysis /
- Beddington-DeAngelis functional response /
- distributed intracellular delays /
- uniformly persistent
MSC: 34D23, 34D20

FullText(HTML)

References

[1]	E. AvilaAbraham, C. PechErika and R. Esquivel. Global stability of a distributed delayed viral model with general incidence rate, Open Mathematics, 2018, 16, 1374-1389. doi: 10.1515/math-2018-0117 CrossRef Google Scholar
[2]	T. Burton and V. Huston, Repellers in systems with infinite delay. Journal of Mathematical Analysis and Applications, 1989, 137, 240-263. doi: 10.1016/0022-247X(89)90287-4 CrossRef Google Scholar
[3]	G. Butler, H. I. Freedman and P. Waltman, Uniform persistence system, Proc. Amer. Math. Soc, 1986, 96, 425-430. doi: 10.1090/S0002-9939-1986-0822433-4 CrossRef Google Scholar
[4]	A. M. Elaiw and S. A. Azoz, Global properties of a class of HIV infection models with Beddington-DeAngelis functional response, Mathematical Methods in the Applied Sciences, 2013, 4, 383-394. Google Scholar
[5]	P. Georgescu and Y. H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal, SIAM Journal on Applied Mathematics, 2006, 67, 337-353. Google Scholar
[6]	J. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM Journal on Mathematical Analysis, 1989, 20, 388-395. doi: 10.1137/0520025 CrossRef Google Scholar
[7]	G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 2009, 22, 1690-1693. doi: 10.1016/j.aml.2009.06.004 CrossRef Google Scholar
[8]	G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections, SIAM Journal on Applied Mathematics, 2010, 70, 2693–2708. Google Scholar
[9]	Y. Kuang, Delay differential equations with applications in population dynamics, Academics Press, Boston, 1993. Google Scholar
[10]	A. Li, Y. Song and D. Xu, Dynamical behavior of a predator-prey system with two delays and stage structure for the prey, Nonlinear Dynamics, 2016, 5, 1-17. doi: 10.5890/JAND.2016.03.001 CrossRef Google Scholar
[11]	M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections, SIAM Journal on Applied Mathematics, 2010, 70, 2434-2448. doi: 10.1137/090779322 CrossRef Google Scholar
[12]	M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, Journal of Mathematical Analysis and Applications, 2010, 361, 38-47. doi: 10.1016/j.jmaa.2009.09.017 CrossRef Google Scholar
[13]	S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Mathematical Biosciences and Engineering, 2010, 3, 675-685. Google Scholar
[14]	Y. Liu and C. Wu, Global dynamics for an HIV infection model with Crowley-Martin functional response and two distributed delays, Journal of Systems Science and Complexity, 2018, 31, 385-395. doi: 10.1007/s11424-017-6038-3 CrossRef Google Scholar
[15]	C. Lv, L. Huang and Z. Yuan, Global stability for an HIV-1 infection model with Beddington-DeAngelis incidence rate and CTL immune response, Commun Nonlinear Sci Numer Simulat, 2014, 19, 121-127. doi: 10.1016/j.cnsns.2013.06.025 CrossRef Google Scholar
[16]	C. C. McCluskey, Global stability fo ran SIR epidemic model with delay and nonlinear incidence, Nonlinear Analysis: Real World Applications, 2010, 11, 3106-3109. doi: 10.1016/j.nonrwa.2009.11.005 CrossRef Google Scholar
[17]	Y. Nakata, Global dynamics of a cell mediated immunity in viral infection models with distributed delays, Journal of Mathematical Analysis and Applications, 2011, 375, 14-27. doi: 10.1016/j.jmaa.2010.08.025 CrossRef Google Scholar
[18]	P. W. Nelson and A. S. Perelson, Mathematical analysis of delay differential equation models of HIV-1 infection, Mathematical Biosciences, 2002, 179, 73-94. doi: 10.1016/S0025-5564(02)00099-8 CrossRef Google Scholar
[19]	M. A. Nowak and C. R. Bangham, Population dynamics of immune responses to persistent viruses, Science, 1996, 272, 74-79. doi: 10.1126/science.272.5258.74 CrossRef Google Scholar
[20]	S. Pankavich, The effects of latent infection on the dynamics of HIV, Differential Equations and Dynamical Systems, 2015, 1, 1-23. Google Scholar
[21]	K. A. Pawelek, S. Liu, F. Pahlevani and L. Rong, A model of HIV-1 infection with two time delays: Mathematical analysis and comparison with patient data, Mathematical Biosciences, 2012, 235, 98-109. doi: 10.1016/j.mbs.2011.11.002 CrossRef Google Scholar
[22]	G. Rost and J. Wu, SEIR epidemiological model with varying infectivity and infinite delay, Mathematical Biosciences and Engineering, 2008, 5, 389-402. doi: 10.3934/mbe.2008.5.389 CrossRef Google Scholar
[23]	A. M. Shehata, A. M. Elaiw, E. K. Elnahary, M. Abul-Ez, Stability analysis of humoral immunity HIV infection models with RTI and discrete delays, International Journal of Dynamics and Control, 2017, 5, 811-831. doi: 10.1007/s40435-016-0235-0 CrossRef Google Scholar
[24]	H. Shu, L. Wang and J. Watmough, Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM Journal on Applied Mathematics, 2013, 73, 1280-13020. doi: 10.1137/120896463 CrossRef Google Scholar
[25]	J. Wang, G. Huang and Y. Takeuchi, Global asymptotic stability for HIV-1 dynamics with two distributed delays, Mathematical Medicine and Biology, 2012, 29, 283-300. doi: 10.1093/imammb/dqr009 CrossRef Google Scholar
[26]	L. Wang and M. Y. Li, Mathematical analysis of the global dynamics of a model for HIV infection of CD4 T cells, Mathematical Biosciences, 2006, 200, 44-57. doi: 10.1016/j.mbs.2005.12.026 CrossRef Google Scholar
[27]	X. Wang, Y. Tao and X. Song, A delayed HIV-1 infection model with Beddington-DeAngelis functional response, Nonlinear Dynamics, 2010, 62, 67-72. doi: 10.1007/s11071-010-9699-1 CrossRef Google Scholar
[28]	H. Xiang, L. Feng and H. Huo, Stability of the virus dynamics model with Beddington-DeAngelis functional response and delays, Applied Mathematical Modelling, 2013, 37, 5414-5423. doi: 10.1016/j.apm.2012.10.033 CrossRef Google Scholar
[29]	H. Zhu and X. Zou, Impact of delays in cell infection and virus production on HIV-1 dynamics, Mathematical Medicine and Biology, 2008, 25, 99-112. doi: 10.1093/imammb/dqm010 CrossRef Google Scholar